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Neutrosophic Sets and Systems, Vol. 33, 2020 University of New Mexico

Alghamdi and Alshehri, Neutrosophic Fuzzy Soft BCK-submodules

Neutrosophic Fuzzy Soft BCK-submodules

R. S. Alghamdi 1,* and N. O. Alshehri 2

1 Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. 2 Department of Mathematics, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah 21589, Saudi Arabia;

[email protected]

* Correspondence: [email protected]

Abstract: The target of this study is to apply the notion of neutrosophic soft sets to the theory of

BCK-modules by introducing the notion of neutrosophic fuzzy soft BCK-submodules and deriving

their basic properties. Also,(𝛼, 𝛽, 𝛾)-soft top of neutrosophic fuzzy soft sets in BCK-modules is

presented. The concept of Cartesian product of neutrosophic fuzzy soft BCK-submodules is defined

and some results are investigated. Finally, an application of neutrosophic fuzzy soft sets in decision

making is investigated and an example demonstrating the successfully application of this method

is provided.

Keywords: BCK-algebras; BCK-modules; soft sets, fuzzy soft sets, neutrosophic sets; neutrosophic

soft sets; neutrosophic fuzzy soft BCK-submodules.

1. Introduction

A soft set theory as a new mathematical tool for dealing with uncertainties was proposed by

Molodtsov in 1999 [21]. He pointed out several directions for the applications of soft sets. In 2002,

Maji et al. [19] described the application of soft set theory to a decision-making problem. They [18]

also studied several operations on the theory of soft sets. Few years later, Chen et al. [11] presented a

new definition of soft set parametrization reduction and compared this definition to the related

concept of attributes reduction in rough set theory. At present, works on the soft set theory are

progressing rapidly. The algebraic structure of set theories dealing with uncertainties has been

studied by some authors. The most appropriate theory for dealing with vagueness is the theory of

fuzzy sets developed by Zadeh [34]. Since then it has become a vigorous area of research in different

domains such as engineering, medical science, social science, physics, statistics, graph theory,

artificial intelligence, signal processing, multiagent systems, pattern recognition, robotics, computer

networks, expert systems, decision making and automata theory.

Neutrosophic set theory was introduced by F. Smarandache in 1998 [28]. It is considered as a

generalization of the fuzzy set. For the first time V.Kandasamy and F. Smarandache [14] introduced

the concept of algebraic structures which has caused a pattern shift in the study of algebraic

structures. Maji [17] had combined the neutrosophic sets with soft sets and introduced a new

mathematical model neutrosophic soft set. The neutrosophic sets aims to model vagueness and

ambiguity in complex system. In recent years, it is applied by many researchers in various fields such

as group of decision making [3, 22], Project scheduling [1,2] and image processing [26, 33] etc.

In 1994, the notion of BCK-modules was introduced by H. Abujable, M. Aslam and A. Thaheem

as an action of BCK-algebras on abelian group [4]. BCK-modules theory then was developed by Z.

perveen, M. Aslam and A. Thaheem [25]. Bakhshi [8] presented the concept of fuzzy BCK-

submodules and investigated their properties. Recently, H. Bashir and Z. Zahid applied the theory

of soft sets on BCK-modules in [16].

Neutrosophic Sets and Systems, Vol. 33, 2020 146

Alghamdi and Alshehri, Neutrosophic Fuzzy Soft BCK-submodules.

In this paper, the concept of neutrosophic fuzzy soft BCK-submodules of BCK-algebra will be

introduced and some related properties will be established. Also, (α,β,γ)-soft top of neutrosophic

fuzzy soft sets in BCK-modules will be presented. We will define the concept of Cartesian product of

neutrosophic fuzzy soft BCK-submodules and investigate some results. Finally, an application of

neutrosophic fuzzy soft sets in decision making is going to be investigated and an example

demonstrating the successfully application of this method will be given.

This paper is classified as follows. Section 2 gives a brief introduction of neutrosophic fuzzy set,

neutrosophic fuzzy soft set, BCK-algebra and BCK-submodule. The notion of neutrosophic fuzzy soft

BCK-submodules and some related results are introduced in section 3. The concept of Cartesian

product of neutrosophic fuzzy soft BCK-submodules and some properties are obtained in section 4.

Section 5 investigates the application of neutrosophic fuzzy soft set in group decision making

problems. Finally, in section 6 conclusion is given.

2. Preliminaries

In this section, some preliminaries from the soft set theory, neutrosophic soft sets, BCK-algebras

and BCK-modules are induced.

Definition 2.1.[21] Let 𝑈 be an initial universe and 𝐸 be a set of parameters. Let 𝑃 (𝑈) denote

the power set of 𝑈 and let 𝐴 be a nonempty subset of 𝐸. A pair 𝐹𝐴 = (𝐹, 𝐴) is called a soft set over

𝑈, where 𝐴 ⊆ 𝐸 and 𝐹 ∶ 𝐴 → 𝑃 (𝑈) is a set-valued mapping, called the approximate function of

the soft set (𝐹, 𝐴). It is easy to represent a soft set (𝐹, 𝐴) by a set of ordered pairs as follows:

(𝐹, 𝐴) = {(𝑥, 𝐹 (𝑥)) ∶ 𝑥 ∈ 𝐴}

Neutrosophic set is a generalization of the fuzzy set especially of intuitionistic fuzzy set. The

intuitionistic fuzzy set has the degree of non-membership as introduced by K. Atanassov [7].

Smarandache in 1998 [28] has introduced the degree of indeterminacy as an independent

component and defined the neutrosophic set on three components: truth, indeterminacy and falsity.

Definition 2.2.[28] A neutrosophic set 𝐴 on the universe of discourse 𝑈 is defined as 𝐴 =

{(𝑥, 𝑇𝐴(𝑥), 𝐼𝐴(𝑥), 𝐹𝐴(𝑥)), 𝑥 ∈ 𝑈} where 𝑇𝐴 ∶ 𝑈 → ] − 0, 1 + [ is a truth membership function,

𝐼𝐴 ∶ 𝑈 → ] − 0, 1 + [ is an indeterminate membership function, and 𝐹𝐴 ∶ 𝑈 → ] − 0, 1 + [ is a false

membership function and 0− ≤ 𝑇𝐴(𝑥) + 𝐼𝐴(𝑥) + 𝐹𝐴(𝑥) ≤ 3 +.

From philosophical point of view, the neutrosophic set takes the value from real standard or

nonstandard subsets of ] − 0, 1 + [. But in real life application in scientific and engineering

problems it is difficult to use neutrosophic set with value from real standard or non-standard subset

of ] − 0, 1 + [. Hence, we consider the neutrosophic set which takes the value from the subset of

[0, 1].

Definition 2.3.[17] Let 𝑈 be an initial universe set and 𝐸 be a set of parameters. Consider 𝐴 ⊂

𝐸. Let 𝑃 (𝑈) denotes the set of all neutrosophic sets of 𝑈. The collection (𝐹, 𝐴) is termed to be the

neutrosophic soft set (NSS) over 𝑈, where 𝐹 is a mapping given by 𝐹 ∶ 𝐴 → 𝑃 (𝑈).

Definition 2.4.[17] Let (𝐹, 𝐴) and (𝐺, 𝐵) be two neutrosophic soft sets over the common

universe 𝑈 . (𝐹, 𝐴) is said to be neutrosophic soft subset of (𝐺, 𝐵) if 𝐴 ⊂ 𝐵 , and 𝑇𝐹(𝑒)(𝑥) ≤

𝑇𝐺(𝑒) (𝑥), 𝐼𝐹(𝑒) (𝑥) ≤ 𝐼𝐺(𝑒) (𝑥), 𝐹𝐹(𝑒) (𝑥) ≥ 𝐹𝐺(𝑒) (𝑥), ∀𝑒 ∈ 𝐴, 𝑥 ∈ 𝑈. We denote it by (𝐹, 𝐴) ⊆ (𝐺, 𝐵).

Definition 2.5. [17] The complement of a neutrosophic soft set (𝐹, 𝐴) denoted by (𝐹, 𝐴)𝑐 and

is defined as (𝐹, 𝐴)𝑐 = (𝐹𝑐 , ∼ 𝐴), where 𝐹𝑐: ∼ 𝐴 → 𝑃 (𝑈) is a mapping given by



𝐹𝑐 (𝑒) = ( 𝑇𝐹𝑐(𝑒) = 𝐹𝐹(𝑒), 𝐼𝐹𝑐(𝑒) = 𝐼𝐹(𝑒), 𝐹𝐹𝑐(𝑒) = 𝑇𝐹(𝑒)) for all 𝑒 ∈∼ 𝐴

Definition 2.6.[12,13] An algebra (𝑋,∗, 0) of type (2, 0) is called BCK-algebra if it is satisfying

the following axioms:

(BCK-1) ((𝑥 ∗ 𝑦) ∗ (𝑥 ∗ 𝑧)) ∗ (𝑧 ∗ 𝑦) = 0,

(BCK-2) (𝑥 ∗ (𝑥 ∗ 𝑦)) ∗ 𝑦 = 0,

(BCK-3) 𝑥 ∗ 𝑥 = 0,

(BCK-4) 0 ∗ 𝑥 = 0,

(BCK-5) 𝑥 ∗ 𝑦 = 0 𝑎𝑛𝑑 𝑦 ∗ 𝑥 = 0 𝑖𝑚𝑝𝑙𝑦 𝑥 = 𝑦, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥, 𝑦, 𝑧 ∈ 𝑋.

A partial ordering “ ≤ ” is defined on 𝑋 by 𝑥 ≤ 𝑦 ⇔ 𝑥 ∗ 𝑦 = 0. A BCK-algebra 𝑋 is said to

be bounded if there is an element 1 ∈ 𝑋 such that 𝑥 ≤ 1, for all 𝑥 ∈ 𝑋, commutative if it satisfies

the identity 𝑥 ∧ 𝑦 = 𝑦 ∧ 𝑥, where 𝑥 ∧ 𝑦 = 𝑦 ∗ (𝑦 ∗ 𝑥), for all 𝑥, 𝑦 ∈ 𝑋 and implicative if

𝑥 ∗ (𝑦 ∗ 𝑥) = 𝑥, for all 𝑥, 𝑦 ∈ 𝑋.

Definition 2.7.[4] Let 𝑋 be a BCK-algebra. Then by a left 𝑋-module (abbreviated 𝑋-module),

we mean an abelian group 𝑀 with an operation 𝑋 × 𝑀 → 𝑀 with (𝑥, 𝑚) ⟼ 𝑥𝑚 satisfies the

following axioms for all 𝑥, 𝑦 ∈ 𝑋 and 𝑚, 𝑛 ∈ 𝑀:

(i) (𝑥 ∧ 𝑦)𝑚 = 𝑥(𝑦𝑚),

(ii) 𝑥(𝑚 + 𝑛) = 𝑥𝑚 + 𝑥𝑛,

(iii) 0𝑚 = 0.

A subgroup 𝑁 of a BCK-module 𝑀 is called submodule of 𝑀 if 𝑁 is also a BCK-module.

Definition 2.8.[8] Let 𝑋 be a BCK-algebra. A subset 𝑁 of a BCK-module 𝑀 is a BCK-

submodule of 𝑀 if and only if 𝑛1 − 𝑛2 ∈ 𝑁 and 𝑥𝑛 ∈ 𝑁 for all 𝑛, 𝑛1, 𝑛2 ∈ 𝑁 and 𝑥 ∈ 𝑋.

Moreover, if 𝑋 is bounded and 𝑀 satisfies 1𝑚 = 𝑚 , for all 𝑚 ∈ 𝑀 , then 𝑀 is said to be

unitary. A mapping µ ∶ 𝑋 → [0, 1] is called a fuzzy set in a BCK-algebra 𝑋. For any fuzzy set µ in

𝑋 and any 𝑡 ∈ [0, 1], we define set 𝑈(µ; 𝑡) = µ 𝑡 = {𝑥 ∈ 𝑋|µ(𝑥) ≥ 𝑡}, which is called an upper

𝑡-level cut of µ.

Definition 2.9.[8] A fuzzy subset µ of 𝑀 is said to be a fuzzy BCK-submodule if for all

𝑚, 𝑚1, 𝑚2 ∈ 𝑀 and 𝑥 ∈ 𝑋, the following axioms hold:

(FBCKM1) µ(𝑚1 + 𝑚2) ≥ min{µ(𝑚1), µ(𝑚2)},

(FBCKM2) µ(−𝑚) = µ(𝑚),

(FBCKM3) µ(𝑥𝑚) ≥ µ(𝑚).

Definition 2.10.[16] A soft set (𝐹, 𝐴) over a BCK-module 𝑀 is said to be a soft BCK-submodule

over 𝑀 if for all 휀 ∈ 𝐴, 𝐹(휀) is a BCK-submodule of 𝑀.

3. Neutrosophic fuzzy soft BCK-submodules

In this section, we introduce the notion of neutrosophic fuzzy soft BCK-submodules and some

related results.

Definition 3.1. A neutrosophic fuzzy soft set (𝐹, 𝐴) over a BCK-module 𝑀 in a BCK-algebra 𝑋

is said to be a neutrosophic fuzzy soft BCK-submodule over 𝑀 if for all 𝑚, 𝑚1, 𝑚2 ∈ 𝑀 , 𝑥 ∈ 𝑋 and

휀 ∈ 𝐴 the following axioms hold:

(NFSS1) 𝑇𝐹( )(𝑚1 + 𝑚2) ≥ min{𝑇𝐹( )(𝑚1), 𝑇𝐹( )(𝑚2)},

𝐼𝐹( )(𝑚1 + 𝑚2) ≥ min{𝐼𝐹( )(𝑚1), 𝐼𝐹( )(𝑚2)},



𝐹𝐹( )(𝑚1 + 𝑚2) ≤ max{𝐹𝐹( )(𝑚1), 𝐹𝐹( )(𝑚2)},

(NFSS2) 𝑇𝐹( )(−𝑚) = 𝑇𝐹( )(𝑚), 𝐼𝐹( )(−𝑚) = 𝐼𝐹( )(𝑚), 𝐹𝐹( )(−𝑚) = 𝐹𝐹( )(𝑚),

(NFSS3) 𝑇𝐹( )(𝑥𝑚) ≥ 𝑇𝐹( )(𝑚), 𝐼𝐹( )(𝑥𝑚) ≥ 𝐼𝐹( )(𝑚), 𝐹𝐹( )(𝑥𝑚) ≤ 𝐹𝐹( )(𝑚).

Example 3.2. Let 𝑋 = {0, 𝑎, 𝑏, 𝑐, 𝑑, 1} be a set with a binary operation ∗ defined in Table 1,

then (𝑋,∗, 0) forms a bounded commutative, non-implicative BCK-algebra (see [20]). (𝑀, +) forms a

commutative group defined in Table 2 where 𝑀 = {0, 𝑎, 𝑐, 𝑑} be a subset of 𝑋. Consequently, 𝑀

forms an 𝑋-module (see [15]).

Table 1 Table 2 Table 3

Let 𝐴 = {0, 𝑎}. Define a neutrosophic fuzzy soft set (𝐹, 𝐴) over 𝑀 as shown in Table 4.

Table 4

Consequently, a routine calculations shows that (𝐹, 𝐴) forms a neutrosophic fuzzy soft BCK-

submodule over M. Note that, Table 3 explains the action of 𝑋 on 𝑀 under the operation 𝑥𝑚 = 𝑥 ∧

𝑚 for all 𝑥 ∈ 𝑋 and 𝑚 ∈ 𝑀.

For the sake of simplicity, we shall use the symbol 𝑁𝐹𝑆𝑆(𝑀) for the set of all neutrosophic fuzzy

soft BCK-submodules over 𝑀.

Theorem 3.3. Let 𝑋 be a BCK-algebra then a neutrosophic fuzzy soft set (𝐹, 𝐴) ∈ 𝑁𝐹𝑆𝑆(𝑀) if

and only if

(i) 𝑇𝐹( )(𝑥𝑚) ≥ 𝑇𝐹( )(𝑚), 𝐼𝐹( )(𝑥𝑚) ≥ 𝐼𝐹( )(𝑚), 𝐹𝐹( )(𝑥𝑚) ≤ 𝐹𝐹( )(𝑚),

(ii) 𝑇𝐹( )(𝑚1 − 𝑚2) ≥ min{𝑇𝐹( )(𝑚1), 𝑇𝐹( )(𝑚2)},

𝐼𝐹( )(𝑚1 − 𝑚2) ≥ min{𝐼𝐹( )(𝑚1), 𝐼𝐹( )(𝑚2)},

𝐹𝐹( )(𝑚1 − 𝑚2) ≤ max{𝐹𝐹( )(𝑚1), 𝐹𝐹( )(𝑚2)}.

for all 𝑚, 𝑚1, 𝑚2 ∈ 𝑀 , 𝑥 ∈ 𝑋 and 휀 ∈ 𝐴.

Proof. Let (𝐹, 𝐴) be a neutrosophic fuzzy soft BCK-submodule over 𝑀 then by the Definition (3.1)

∧ 0 𝑎 𝑐 𝑑 0 0 0 0 0 𝑎 0 𝑎 0 𝑎 𝑏 0 𝑎 0 𝑎 𝑐 0 0 𝑐 𝑐 𝑑 0 𝑎 𝑐 𝑑 1 0 𝑎 𝑐 𝑑

∗ 0 𝑎 𝑏 𝑐 𝑑 1 0 0 0 0 0 0 0 𝑎 𝑎 0 0 𝑎 0 0 𝑏 𝑏 𝑎 0 𝑏 𝑎 0 𝑐 𝑐 𝑐 𝑐 0 0 0 𝑑 𝑑 𝑐 𝑐 𝑎 0 0 1 1 𝑑 𝑐 𝑏 𝑎 0

+ 0 𝑎 𝑐 𝑑 0 0 a c d 𝑎 𝑎 0 𝑑 𝑐 𝑐 𝑐 𝑑 0 𝑎 𝑑 𝑑 𝑐 𝑎 0

(𝐹, 𝐴) 0 𝑎 𝑐 𝑑 𝑇𝐹(0) 0.9 0.7 0.8 0.7

𝐼𝐹(0) 0.8 0.5 0.6 0.5

𝐹𝐹(0) 0.1 0.1 0.1 0.1

𝑇𝐹(𝑎) 0.5 0.2 0.3 0.2

𝐼𝐹(𝑎) 0.3 0.1 0.3 0.1

𝐹𝐹(𝑎) 0.1 0.5 0.4 0.5



condition (i) holds.

(ii) 𝑇𝐹( ) (𝑚1 − 𝑚2) = 𝑇𝐹( ) (𝑚1 + (−𝑚2)) ≥ min{𝑇𝐹( ) (𝑚1), 𝑇𝐹( ) (−𝑚2)} = min{𝑇𝐹( ) (𝑚1), 𝑇𝐹( ) (𝑚2)},

Similarly for

𝐼𝐹( )(𝑚1 − 𝑚2) ≥ min{𝐼𝐹( )(𝑚1), 𝐼𝐹( )(𝑚2)}

and

𝐹𝐹( )(𝑚1 − 𝑚2) ≤ max{𝐹𝐹( )(𝑚1), 𝐹𝐹( )(𝑚2)}.

Conversely suppose (𝐹, 𝐴) satisfies the conditions (i), (ii). Then we have by (i)

𝑇𝐹( )(−𝑚) = 𝑇𝐹( )((−1)𝑚) ≥ 𝑇𝐹( )(𝑚), and 𝑇𝐹( )(𝑚) = 𝑇𝐹( )((−1)(−1)𝑚) ≥ 𝑇𝐹( )(−𝑚).

Thus, 𝑇𝐹( )(𝑚) = 𝑇𝐹( )(−𝑚). Similarly for 𝐼𝐹( )(−𝑚) = 𝐼𝐹( ) (𝑚) 𝑎𝑛𝑑 𝐹𝐹( )(−𝑚) = 𝐹𝐹( )(𝑚).

(ii)𝑇𝐹( ) (𝑚1 + 𝑚2) = 𝑇𝐹( ) (𝑚1 − (−𝑚2)) ≥ min{𝑇𝐹( ) (𝑚1), 𝑇𝐹( ) (−𝑚2)} = min{𝑇𝐹( ) (𝑚1), 𝑇𝐹( ) (𝑚2)},

Similarly for

𝐼𝐹( )(𝑚1 + 𝑚2) ≥ min{𝐼𝐹( )(𝑚1), 𝐼𝐹( )(𝑚2)}

and

𝐹𝐹( )(𝑚1 + 𝑚2) ≤ max{𝐹𝐹( )(𝑚1), 𝐹𝐹( )(𝑚2)}.

Hence (𝐹, 𝐴) is a neutrosophic fuzzy soft BCK-submodule over 𝑀.

Theorem 3.4. A neutrosophic fuzzy soft set (𝐹, 𝐴) belongs to 𝑁𝐹𝑆𝑆(𝑀) in a BCK-algebra 𝑋 if and

only if for all 𝑚, 𝑚1, 𝑚2 ∈ 𝑀 , 𝑥, 𝑦 ∈ 𝑋 𝑎𝑛𝑑 휀 ∈ 𝐴 the following statements hold:

(i) 𝑇𝐹( )(0) ≥ 𝑇𝐹( )(𝑚), 𝐼𝐹( )(0) ≥ 𝐼𝐹( )(𝑚), 𝐹𝐹( )(0) ≤ 𝐹𝐹( )(𝑚),

(ii) 𝑇𝐹( )(𝑥𝑚1 − 𝑦𝑚2) ≥ min{𝑇𝐹( )(𝑚1), 𝑇𝐹( )(𝑚2)},

𝐼𝐹( )(𝑥𝑚1 − 𝑦𝑚2) ≥ min{𝐼𝐹( )(𝑚1), 𝐼𝐹( )(𝑚2)},

𝐹𝐹( )(𝑥𝑚1 − 𝑦𝑚2) ≤ max{𝐹𝐹( )(𝑚1), 𝐹𝐹( )(𝑚2)}.

Proof. Let (𝐹, 𝐴) be a 𝑁𝐹𝑆𝑆(𝑀), by Theorem (3.3) and since 0𝑚 = 0 for all 𝑚 ∈ 𝑀, we have

(i) 𝑇𝐹(휀)(0) = 𝑇𝐹(휀)(0𝑚) ≥ 𝑇𝐹(휀)(𝑚).

The same way for 𝐼𝐹( )(0) ≥ 𝐼𝐹( )(𝑚) and 𝐹𝐹( )(0) ≤ 𝐹𝐹( )(𝑚).

(ii) 𝑇𝐹( )(𝑥𝑚1 − 𝑦𝑚2) ≥ min{𝑇𝐹( )(𝑥𝑚1), 𝑇𝐹( )(𝑦𝑚2)} ≥ min{𝑇𝐹( ) (𝑚1), 𝑇𝐹( ) (𝑚2)}.

Similarly for

𝐼𝐹( )(𝑥𝑚1 − 𝑦𝑚2) ≥ min{𝐼𝐹( )(𝑚1), 𝐼𝐹( )(𝑚2)}.

and

𝐹𝐹( )(𝑥𝑚1 − 𝑦𝑚2) ≤ max{𝐹𝐹( )(𝑚1), 𝐹𝐹( )(𝑚2)}.

Conversely suppose (𝐹, 𝐴) satisfies (i), (ii), then

𝑇𝐹( )(𝑥𝑚 ) = 𝑇𝐹( )( 𝑥(𝑚 − 0)) = 𝑇𝐹( )( 𝑥𝑚 − 𝑥0) ≥ min{𝑇𝐹( )(𝑚), 𝑇𝐹( )(0)} = 𝑇𝐹( )(𝑚).

Similarly for 𝐼𝐹( )(𝑥𝑚) ≥ 𝐼𝐹( )(𝑚) and 𝐹𝐹( )(𝑥𝑚) ≤ 𝐹𝐹( )(𝑚).

Also,

𝑇𝐹( )(𝑚1 − 𝑚2) = 𝑇𝐹( )(1𝑚1 − 1𝑚2) ≥ min{𝑇𝐹( )(𝑚1), 𝑇𝐹( )(𝑚2)}.

Similarly for

𝐼𝐹( )(𝑚1 − 𝑚2) ≥ min{𝐼𝐹( )(𝑚1), 𝐼𝐹( )(𝑚2)}

and



𝐹𝐹( )(𝑚1 − 𝑚2) ≤ max{𝐹𝐹( )(𝑚1), 𝐹𝐹( )(𝑚2)}

Hence by Theorem (3.3), (𝐹, 𝐴) is a neutrosophic fuzzy soft BCK-submodule over M.

Definition 3.5. Let (𝐹, 𝐴) be a neutrosophic fuzzy soft set over 𝑀. Then (𝛼, 𝛽, 𝛾)-soft top of (𝐹, 𝐴)

is a soft set given by (𝐻, 𝐶(𝛼,𝛽,𝛾)(𝐴)) = ((𝑇)𝛼 , (𝐼)𝛽 , (𝐹)𝛾) where

𝐻 (𝑎) = { 𝑚 ∈ 𝑀 ∶ 𝑇𝐹( )(𝑚) ≥ 𝛼, 𝐼𝐹( ) (𝑚) ≥ 𝛽, 𝐹𝐹( )(𝑚) ≤ 𝛾}

for all 휀 ∈ 𝐴, 𝑎 ∈ 𝐶(𝛼,𝛽,𝛾)(𝐴) and 𝛼, 𝛽, 𝛾 ∈ [0, 1] with 𝛼 + 𝛽 + 𝛾 ≤ 3.

Proposition 3.6. A soft set over BCK-module 𝑀 is a neutrosophic fuzzy soft BCK-submodule

over 𝑀 if and only if the (𝛼, 𝛽, 𝛾)-soft top is either empty or soft BCK-submodule over 𝑀 for all

𝛼, 𝛽 , 𝛾 ∈ [0, 1] with 𝛼 + 𝛽 + 𝛾 ≤ 3.

Proof. Let (𝐹, 𝐴) be a 𝑁𝐹𝑆𝑆(𝑀), (𝐻, 𝐶(𝛼,𝛽,𝛾) (𝐴)) is non-empty (𝛼, 𝛽, 𝛾)-soft top of (𝐹, 𝐴) and 𝑋 is a

BCK-algebra. Let 𝑚, 𝑛 ∈ 𝐻 (𝑎) then by Definition (3.5) we have

𝑇𝐹( )(𝑚) ≥ 𝛼, 𝑇𝐹( )(𝑛) ≥ 𝛼 ⇒ min{ 𝑇𝐹( ) (𝑚), 𝑇𝐹( ) (𝑛)} ≥ 𝛼,

𝐼𝐹( )(𝑚) ≥ 𝛽, 𝐼𝐹( )(𝑛) ≥ 𝛽 ⇒ min{𝐼𝐹( )(𝑚), 𝐼𝐹( )(𝑛)} ≥ 𝛽,

𝐹𝐹( )(𝑚) ≤ 𝛾, 𝐹𝐹( )(𝑛) ≤ 𝛾 ⇒ max{ 𝐹𝐹( )(𝑚), 𝐹𝐹( )(𝑛)} ≤ 𝛾.

By Theorem (3.3), we have

𝑇𝐹( )(𝑚 − 𝑛) ≥ min{ 𝑇𝐹( )(𝑚), 𝑇𝐹( )(𝑛)} ≥ 𝛼,

𝐼𝐹( )(𝑚 − 𝑛) ≥ min{𝐼𝐹( )(𝑚), 𝐼𝐹( )(𝑛)} ≥ 𝛽,

𝐹𝐹( )(𝑚 − 𝑛) ≤ max{ 𝐹𝐹( )(𝑚), 𝐹𝐹( )(𝑛)} ≤ 𝛾.

Hence 𝑚 − 𝑛 ∈ 𝐻 (𝑎).

Now let 𝑚 ∈ 𝐻 (𝑎), 𝑥 ∈ 𝑋. Then

𝑇𝐹( )(𝑥𝑚) ≥ 𝑇𝐹( )(𝑚) ≥ 𝛼,

𝐼𝐹( )(𝑥𝑚) ≥ 𝐼𝐹( )(𝑚) ≥ 𝛽 ,

𝐹𝐹( )(𝑥𝑚) ≤ 𝐹𝐹( )(𝑚) ≤ 𝛾.

Hence 𝑥𝑚 ∈ 𝐻 (𝑎). Therfore 𝐻 (𝑎) is a BCK-submodule of 𝑀 and (𝐻, 𝐶(𝛼,𝛽,𝛾) (𝐴)) is a soft BCK-

submodule over 𝑀.

Conversely, let (𝐻, 𝐶(𝛼,𝛽,𝛾) (𝐴)) is a soft BCK-submodule over 𝑀 for all 𝛼, 𝛽, 𝛾 ∈ [0, 1] with 𝛼 + 𝛽 +

𝛾 ≤ 3. Let𝛼 = min{ 𝑇𝐹( )(𝑚), 𝑇𝐹( )(𝑛)}, 𝛽 = min{ 𝐼𝐹( )(𝑚), 𝐼𝐹( )(𝑛)} and 𝛾 = max{ 𝐹𝐹( )(𝑚), 𝐹𝐹( )(𝑛)}

for 𝑚, 𝑛 ∈ 𝑀. Then 𝑚, 𝑛 ∈ 𝐻 (𝑎). Since 𝐻(𝑎)is a BCK-submodule of 𝑀, therefore 𝑚 − 𝑛 ∈ 𝐻 (𝑎)

which mean

𝑇𝐹( ) (𝑚 − 𝑛) ≥ 𝛼 = min{ 𝑇𝐹( ) (𝑚), 𝑇𝐹( ) (𝑛)} ,

𝐼𝐹( ) (𝑚 − 𝑛) ≥ 𝛽 = min{𝐼𝐹( ) (𝑚), 𝐼𝐹( ) (𝑛)} ,

𝐹𝐹( )(𝑚 − 𝑛) ≤ 𝛾 = max{ 𝐹𝐹( ) (𝑚), 𝐹𝐹( ) (𝑛)} .

Now let 𝛼 = 𝑇𝐹( )(𝑚) , 𝛽 = 𝐼𝐹( )(𝑚) and 𝛾 = 𝐹𝐹( )(𝑚) then 𝑚 ∈ 𝐻 (𝑎) . Since 𝐻 (𝑎) is a BCK-

submodule of 𝑀 then 𝑥𝑚 ∈ 𝐻 (𝑎) for all 𝑥 ∈ 𝑋 i.e.

𝑇𝐹( )(𝑥𝑚) ≥ 𝛼 = 𝑇𝐹( )(𝑚), 𝐼𝐹( )(𝑥𝑚) ≥ 𝛽 = 𝐼𝐹( )(𝑚) and 𝐹𝐹( )(𝑥𝑚) ≤ 𝛾 = 𝐹𝐹( )(𝑚). By Theorem

(3.3) we have, (𝐹, 𝐴) is a neutrosophic fuzzy soft BCK-submodule over 𝑀.

Definition 3.7. Let (𝐹, 𝐴) be a neutrosophic fuzzy soft set over 𝑀, then ( �̃� , 𝐴01 ) is called soft

support of (𝐹, 𝐴) if it satisfies

�̃� (𝛿) = { 𝑚 ∈ 𝑀 ∶ 𝑇𝐹( ) (𝑚) > 0, 𝐼𝐹( ) (𝑚) > 0, 𝐹𝐹( ) (𝑚) < 1}

for all 𝛿 ∈ 𝐴01 and 𝑚 ∈ 𝑀.



Theorem 3.8. Let (𝐹, 𝐴) be a neutrosophic fuzzy soft BCK-submodule over 𝑀, then ( �̃� , 𝐴01 ) is

a soft BCK-submodule over 𝑀.

Proof. Let (𝐹, 𝐴) be a neutrosophic fuzzy soft BCK-submodule over 𝑀 in a BCK-algebra 𝑋 and let

𝑚1, 𝑚2 ∈ �̃�(𝑎), 𝑎 ∈ 𝐴01 then

𝑇𝐹( )(𝑚1 − 𝑚2) ≥ min{ 𝑇𝐹( )(𝑚1), 𝑇𝐹( )(𝑚2)} > 0, 𝐼𝐹( )(𝑚1 − 𝑚2) ≥ min{ 𝐼𝐹( )(𝑚1), 𝐼𝐹( )(𝑚2)} > 0,

and 𝐹𝐹( )(𝑚1 − 𝑚2) ≤ max{ 𝐹𝐹( )(𝑚1), 𝐹𝐹( )(𝑚2)} < 1. So, 𝑚1 − 𝑚2 ∈ �̃�(𝑎).

Now let 𝑚 ∈ �̃�(𝑎), 𝑥 ∈ 𝑋, then we have 𝑇𝐹( ) (𝑥𝑚) ≥ 𝑇𝐹( ) (𝑚) > 0, 𝐼𝐹( ) (𝑥𝑚) ≥ 𝐼𝐹( ) (𝑚) > 0,

and 𝐹𝐹( ) (𝑥𝑚) ≤ 𝐹𝐹( ) (𝑚) < 1. So, 𝑥𝑚 ∈ �̃�(𝑎). Hence �̃�(𝑎) is a BCK-submodule of 𝑀. Therefore

( �̃� , 𝐴01 ) a soft BCK-submodule over 𝑀.

Proposition 3.9 If a neutrosophic fuzzy soft set over 𝑀 is a neutrosophic fuzzy soft BCK-

submodule over M, then the complement of a neutrosophic fuzzy soft set is also neutrosophic fuzzy

soft BCK-submodule over 𝑀.

Proof. The proof follow from the Theorem (3.3) and Definition (2.5).

Corollary 3.10 Let (𝐹, 𝐴) be a neutrosophic fuzzy soft BCK-submodule over 𝑀 if and only if

(𝐹, 𝐴)𝑐 is a neutrosophic fuzzy soft BCK-submodule over 𝑀.

4. Cartesian Product of Neutrosophic Fuzzy Soft BCK-submodules

In this section, we defined the concept of Cartesian product of neutrosophic fuzzy soft BCK-

submodules and obtained some properties on it.

Definition 4.1. Let (𝐹, 𝐴) and (𝐺, 𝐵) be two neutrosophic fuzzy soft BCK-submodules over 𝑀.

Then the Cartesian product (𝐹, 𝐴) × (𝐺, 𝐵) = (𝐻, 𝐶) where 𝐶 = 𝐴 × 𝐵 and 𝐻 (휀, 𝛿) = 𝐹 (휀) ×

𝐺(𝛿) for all (휀, 𝛿) ∈ 𝐴 × 𝐵 defined as 𝐻(휀, 𝛿) = (𝑇𝐹×𝐺(𝑚, 𝑛), 𝐼𝐹×𝐺(𝑚, 𝑛), 𝐹𝐹×𝐺(𝑚, 𝑛)) where

𝑇𝐻( ,𝛿)(𝑚, 𝑛) = 𝑇𝐹×𝐺 (𝑚, 𝑛) = min{ 𝑇𝐹( ) (𝑚), 𝑇𝐺(𝛿) (𝑛)} ,

𝐼𝐻( ,𝛿)(𝑚, 𝑛) = 𝐼𝐹×𝐺 (𝑚, 𝑛) = min{ 𝐼𝐹( ) (𝑚), 𝐼𝐺(𝛿) (𝑛)} ,

𝐹𝐻( ,𝛿)(𝑚, 𝑛) = 𝐹𝐹×𝐺 (𝑚, 𝑛) = max{ 𝐹𝐹( ) (𝑚), 𝐹𝐺(𝛿) (𝑛)} .

For all 𝑚, 𝑛 ∈ 𝑀 and 𝑇𝐻 , 𝐼𝐻 , 𝐹𝐻: 𝑀 × 𝑀 → [0,1].

Theorem 4.2. Let (𝐹, 𝐴) and (𝐺, 𝐵) be two neutrosophic fuzzy soft BCK-submodules over 𝑀.

Then (𝐹, 𝐴) × (𝐺, 𝐵) is a neutrosophic fuzzy soft BCK-submodule over 𝑀 × 𝑀.

Proof. Let 𝑋 be a BCK-algebra and (𝐹, 𝐴), (𝐺, 𝐵) be a neutrosophic fuzzy soft BCK-submodules

over 𝑀. Let 𝑚 ∈ 𝑀, then by Definition (4.1) and Theorem (3.4)

𝑇𝐹×𝐺 (0,0) = min{ 𝑇𝐹( ) (0), 𝑇𝐺(𝛿) (0)} ≥ min{𝑇𝐹( ) (𝑚), 𝑇𝐺(𝛿) (𝑚)} = 𝑇𝐹×𝐺 (𝑚, 𝑚),

The same for 𝐼𝐹×𝐺 (0,0) ≥ 𝐼𝐹×𝐺 (𝑚, 𝑚) and 𝐹𝐹×𝐺 (0,0) ≤ 𝐹𝐹×𝐺 (𝑚, 𝑚) for all (휀, 𝛿) ∈ 𝐴 × 𝐵.

Also, for any (𝑚1, 𝑛1), (𝑚2, 𝑛2) ∈ 𝑀 × 𝑀 and 𝑥, 𝑦 ∈ 𝑋 we have

𝑇𝐹×𝐺(𝑥𝑚1 − 𝑦𝑚2, 𝑥𝑛1 − 𝑦𝑛2) = min{ 𝑇𝐹( )(𝑥𝑚1 − 𝑦𝑚2), 𝑇𝐺(𝛿)(𝑥𝑛1 − 𝑦𝑛2)}

≥ min{min{𝑇𝐹( )(𝑚1), 𝑇𝐹( )(𝑚2)} , min{𝑇𝐺(𝛿)(𝑛1), 𝑇𝐺(𝛿)(𝑛2)}}

= min{min{𝑇𝐹( )(𝑚1), 𝑇𝐺(𝛿)(𝑛1)} , min{𝑇𝐹( )(𝑚2), 𝑇𝐺(𝛿)(𝑛2)}}

= min{𝑇𝐹×𝐺(𝑚1, 𝑛1), 𝑇𝐹×𝐺(𝑚2, 𝑛2)}.

Similarly for

𝐼𝐹×𝐺(𝑥𝑚1 − 𝑦𝑚2, 𝑥𝑛1 − 𝑦𝑛2) ≥ min{𝐼𝐹×𝐺(𝑚1, 𝑛1), 𝐼𝐹×𝐺(𝑚2, 𝑛2)}

and

𝐹𝐹×𝐺(𝑥𝑚1 − 𝑦𝑚2, 𝑥𝑛1 − 𝑦𝑛2) ≤ max{𝐹𝐹×𝐺(𝑚1, 𝑛1), 𝐹𝐹×𝐺(𝑚2, 𝑛2)}



Hence (𝐹, 𝐴) × (𝐺, 𝐵) is a neutrosophic fuzzy soft BCK-submodule over 𝑀 × 𝑀 . The converse of

Theorem 4.2 is not true in general as seen in the following example.

Example 4.3. Let 𝑋 = {0, 𝑎, 𝑏, 𝑐} with a binary operation ∗ defined in Table 5, and then

(𝑋,∗, 0) forms a bounded implicative BCK-algebra (see [20]). Let 𝑀 = {0, 𝑎} be a subset of 𝑋 with a

binary operation + defined by Table 6. Then 𝑀 is a commutative group. Define scalar

multiplication (𝑋, 𝑀) → 𝑀 by 𝑥𝑚 = 𝑥 ∧ 𝑚 for all 𝑥 ∈ 𝑋 and 𝑚 ∈ 𝑀 that is given in Table 7.

Consequently, 𝑀 forms an 𝑋-module (see [15]).

Table 5 Table 6 Table 7

Let 𝐴 = 𝐵 = 𝑀 . Then 𝐶 = 𝐴 × 𝐵 = {(0, 0), (0, 𝑎), (𝑎, 0), (𝑎, 𝑎)}. Define a neutrosophic fuzzy

soft set (𝐻, 𝐶) on 𝑀 × 𝑀 as shown in Table 8.

Table 8

Then (𝐻, 𝐶) = (𝐹, 𝐴) × (𝐺, 𝐵) is a neutrosophic fuzzy soft BCK-submodule over 𝑀 × 𝑀. But if

we consider the neutrosophic fuzzy soft sets (𝐹, 𝐴) and (𝐺, 𝐵) defined as in Table 9 and Table 10.

Table9 Table 10

we can observe that (𝐺, 𝐵) is not a neutrosophic fuzzy soft BCK-submodule over M since

𝑇𝐺(0) (0𝑎) = 𝑇𝐺(0) (0 ∧ 𝑎) = 𝑇𝐺(0) (0) = 0.3 ≱ 𝑇𝐺(0) (𝑎) = 0.4

Proposition.4.4. Let (𝐹, 𝐴) and (𝐺, 𝐵) be two neutrosophic fuzzy soft BCK-submodules over

𝑀. Then the following equalities are satisfied for the (𝛼, 𝛽, 𝛾)-soft top:

∗ 0 𝑎 𝑏 𝑐 0 0 0 0 0 𝑎 𝑎 0 𝑎 0 𝑏 𝑏 𝑏 0 0 𝑐 𝑐 𝑏 𝑎 0

∧ 0 𝑎 0 0 0 𝑎 0 𝑎 𝑏 0 0 𝑐 0 𝑎

+ 0 𝑎 0 0 𝑎 𝑎 𝑎 0

(𝐻, 𝐶) (0,0) (0, 𝑎) (𝑎, 0) (𝑎, 𝑎) 𝑇𝐻(0,0) 0.3 0.3 0.2 0.2

𝐼𝐻(0,0) 0.7 0.5 0.6 0.5

𝐹𝐻(0,0) 0.1 0.5 0.4 0.5

𝑇𝐻(0,𝑎) 0.1 0.1 0.1 0.1

𝐼𝐻(0,𝑎) 0.1 0.1 0.1 0.1

𝐹𝐻(0,𝑎) 0.5 0.6 0.5 0.6

𝑇𝐻(𝑎,0) 0.2 0.2 0.2 0.2

𝐼𝐻(𝑎,0) 0.1 0.1 0.1 0.1

𝐹𝐻(𝑎,0) 0.4 0.5 0.5 0.5

𝑇𝐻(𝑎,𝑎) 0.1 0.1 0.1 0.1

𝐼𝐻(𝑎,𝑎) 0.1 0.1 0.1 0.1

𝐹𝐻(𝑎,𝑎) 0.5 0.6 0.5 0.6

(𝐹, 𝐴) 0 𝑎 𝑇𝐹(0) 0.3 0.2

𝐼𝐹(0) 0.8 0.6

𝐹𝐹(0) 0.1 0.4

𝑇𝐹(𝑎) 0.2 0.2

𝐼𝐹(𝑎) 0.1 0.1

𝐹𝐹(𝑎) 0.4 0.5

(𝐺, 𝐵) 0 𝑎 𝑇𝐺(0) 0.3 0.4

𝐼𝐺(0) 0.7 0.5

𝐹𝐺(0) 0.1 0.5

𝑇𝐺(𝑎) 0.1 0.1

𝐼𝐺(𝑎) 0.1 0.1

𝐹𝐺(𝑎) 0.5 0.6



(𝑇𝐹×𝐺)𝛼 = (𝑇𝐹( ))𝛼 × (𝑇𝐺(𝛿))𝛼 , (𝐼𝐹×𝐺)𝛽 = (𝐼𝐹( ))𝛽 × (𝐼𝐺(𝛿))𝛽 and (𝐹𝐹×𝐺)𝛾 = (𝐹𝐹( ))𝛾 × (𝐹𝐺(𝛿))𝛾

For all (휀, 𝛿) ∈ 𝐴 × 𝐵.

Proof: Let (𝑥, 𝑦) ∈ (𝑇𝐹×𝐺)𝛼 be arbitrary. So

𝑇𝐹×𝐺(𝑥, 𝑦) ≥ 𝛼 ⇔ min{ 𝑇𝐹( )(𝑥), 𝑇𝐺(𝛿)(𝑦)} ≥ 𝛼

⇔ 𝑇𝐹( )(𝑥) ≥ 𝛼, 𝑇𝐺(𝛿)(𝑦) ≥ 𝛼

⇔ (𝑥, 𝑦) ∈ (𝑇𝐹( ))𝛼 × (𝑇𝐺(𝛿))𝛼 .

(𝐼𝐹×𝐺)𝛽 = (𝐼𝐹( ))𝛽 × (𝐼𝐺(𝛿))𝛽 is proved in similar way. Now let(𝑥, 𝑦) ∈ (𝐹𝐹×𝐺)𝛾. Then

𝐹𝐹×𝐺(𝑥, 𝑦) ≤ 𝛾 ⇔ max{ 𝐹𝐹( )(𝑥), 𝐹𝐺(𝛿)(𝑦)} ≤ 𝛾

⇔ 𝐹𝐹( )(𝑥) ≤ 𝛾, 𝐹𝐺(𝛿)(𝑦) ≤ 𝛾

⇔ (𝑥, 𝑦) ∈ (𝐹𝐹( ))𝛾 × (𝐹𝐺(𝛿))𝛾 .

Hence the equalities (𝑇𝐹×𝐺)𝛼 = (𝑇𝐹( ))𝛼 × (𝑇𝐺(𝛿))𝛼 , (𝐼𝐹×𝐺)𝛽 = (𝐼𝐹( ))𝛽 × (𝐼𝐺(𝛿))𝛽 and (𝐹𝐹×𝐺)𝛾 =

(𝐹𝐹( ))𝛾 × (𝐹𝐺(𝛿))𝛾 are satisfied for all (휀, 𝛿) ∈ 𝐴 × 𝐵.

5. The Neutrosophic Fuzzy Soft Set Application in a Decision-Making Problems

In this section we have investigated the application of neutrosophic fuzzy soft set in group

decision making problems. Let 𝑈 = {𝑢1, 𝑢2, … , 𝑢𝑛} be a universal set consisting set of alternatives.

Let 𝐸 = {𝑒1, 𝑒2, … , 𝑒𝑚} be a set of criteria. We can represent a group decision making problem using

the neutrosophic fuzzy soft approach in the following way.

Let (𝐹, 𝐴) denotes the corresponding neutrosophic fuzzy soft set in which 𝐹(𝑒𝑗) represents the

neutrosophic fuzzy set for the alternative 𝑢𝑖 corresponding to the criteria 𝑒𝑗.

Definition.5.1.[17] Let 𝐴 = ⟨𝑇𝐴, 𝐼𝐴 , 𝐹𝐴⟩ be a neutrosophic fuzzy number, and then the score

function 𝑆(𝐴) is defined as follows

𝑆(𝐴) = (𝑇𝐴 + 1 − 𝐼𝐴 + 1 − 𝐹𝐴)/3

For two neutrosophic fuzzy numbers 𝐴 and 𝐵, if 𝑆(𝐴) > 𝑆(𝐵) then 𝐴 > 𝐵.

Algorithm

Step 1: Input the neutrosophic soft set (𝐹, 𝐴).

Step 2: Compute the score function 𝑆(𝐴) of a neutrosophic fuzzy number 𝐴 = ⟨𝑇𝐴 , 𝐼𝐴, 𝐹𝐴⟩, based on

the truth-membership degree, indeterminacy-membership degree and falsity membership degree

by 𝑆(𝐴) = (𝑇𝐴 + 1 − 𝐼𝐴 + 1 − 𝐹𝐴)/3 and the induced fuzzy soft set Δ �̃� = ( �̃�, 𝐴).

Step 3: Calculate the average of �̃�(𝑒𝑗) for each 𝑢𝑖 and let it be denoted as 𝑎𝑖 , this is the decision

table.

Step 4: Select the optimal alternative 𝑢𝑖 if 𝑎𝑖 = max𝑘

(𝑎𝑘).

Step 5: If there are more than one 𝑢𝑖 's then any one of 𝑢𝑖 may be chosen.

Remark 5.2:

In the case of multicriteria decision making problems, sometimes every criteria 𝑒𝑗 associated

with the value 𝑤𝑗 ∈ [0, 1] called its weight, which used to represent the different importance of the

concerned criteria. In this case there is a small change in the above algorithm. In step 3 instead of

average we take weighted average



∑ �̃�(𝑒𝑗)𝑤𝑗𝑚𝑗=1

𝑚

and follows the next steps.

We adopt the following example to illustrate the idea of algorithm given above.

Example 5.3. Suppose that someone wants to invest his money in a stock exchange company.

Let 𝑈 = {𝑢1, 𝑢2, 𝑢3, 𝑢4} the set of alternative companies. Then the four alternatives are evaluated

over the set of criteria 𝐸 = {𝑒1, 𝑒2, 𝑒3, 𝑒4} where 𝑒1=Earnings Per Share, 𝑒2=Dividend, 𝑒3=Book

Value and 𝑒4=Price/Earning Ratio. The Evaluation values of the four alternatives on the basis of

the above four criteria using the form of neutrosophic fuzzy soft set. The problem is the selection of

best company which satisfies the criteria.

Step 1: Neutrosophic fuzzy soft set (𝐹, 𝐴) can describe in Table 11.

Table 11

Step 2: calculate the score of each neutrosophic fuzzy number and obtain the induced fuzzy soft

set

Δ �̃� = ( �̃�, 𝐴), which is shown in Table 12.

Table 12

Step 3: Calculate the average of �̃�(𝑒𝑗) and the decision table for each company 𝑢𝑖 obtained in

Table 13.

Table 13

Step 4: Rank all the alternative companies according to the average values 𝑎𝑖 (𝑖 = 1, 2, 3, 4) as:

𝑢4 ≻ 𝑢3 ≻ 𝑢1 ≻ 𝑢2

and thus 𝑢4 is the most desirable alternative.

6. Conclusion

In this paper, we introduced the concept of neutrosophic fuzzy soft BCK-submodules of BCK-

algebra and established some related properties. Also, (𝛼, 𝛽, 𝛾)-soft top of neutrosophic fuzzy soft

sets in BCK-modules was presented. We defined the concept of Cartesian product of neutrosophic

fuzzy soft BCK-submodules and investigated some results. Then, we presented an application

method for the neutrosophic fuzzy soft set theory in decision making problem. Finally, we provided

𝐹 𝑒1 𝑒2 𝑒3 𝑒4

𝑢1 ⟨0.4,0.2,0.5⟩ ⟨0.5,0.3,0.3⟩ ⟨0.2,0.7,0.5⟩ ⟨0.4,0.6,0.5⟩

𝑢2 ⟨0.3,0.6,0.1⟩ ⟨0.2,0.6,0.1⟩ ⟨0.4,0.2,0.5⟩ ⟨0.2,0.7,0.5⟩

𝑢3 ⟨0.3,0.5,0.2⟩ ⟨0.4,0.5,0.2⟩ ⟨0.9,0.5,0.7⟩ ⟨0.3,0.7,0.6⟩

𝑢4 ⟨0.6,0.7,0.5⟩ ⟨0.8,0.4,0.6⟩ ⟨0.6,0.3,0.6⟩ ⟨0.8,0.3,0.2⟩

�̃� 𝑒1 𝑒2 𝑒3 𝑒4

𝑢1 0.57 0.63 0.33 0.43

𝑢2 0.53 0.5 0.57 0.33

𝑢3 0.53 0.57 0.57 0.33

𝑢4 0.47 0.6 0.57 0.77

𝑎𝑖 values

𝑎1 0.49

𝑎2 0.4825

𝑎3 0.5

𝑎4 0.6025



an example demonstrating the successfully application of this method. The study of neutrosophic

fuzzy soft set and their properties have a considerable significance in the sense of applications as well

as in understanding the fundamentals of uncertainty. In the future, we shall further develop more

algorithms for neutrosophic fuzzy soft set and apply them to solve practical applications in areas

such as group decision making, image processing, fusion images and so on.

Funding: This research received no external funding.

Conflict of Interest: The authors declare that there is no conflict of interests regarding the publication

of this paper.

Ethical Approval: This article does not contain any studies with human participants or animals

performed by any of the authors.

Data Availability: The data used to support the findings of this study are included within the

article.

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Received: 13, December, 2019. Accepted: May 02, 2020

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